Lectures on Scattering Theory

DMITRI YAFAEV

Lecture notes prepared by Andrew Hassell, based on lectures given by the author at the Australian National University in October and November, 2001.

The ﬁrst two lectures are devoted to describing the basic concepts of scattering theory in a very compressed way. A detailed presentation of the abstract part can be found in [33] and numerous applications in [30] and [36]. The last two lectures are based on the recent research of the author.

1. Introduction to Scattering theory. Trace class method 1. Let us, first, indicate the place of scattering theory amongst other mathematical theories. This is very simple: it is a subset of perturbation theory. The ideology of perturbation theory is as follows. Let H0 and H = H0 +V be self-adjoint operators on a Hilbert space H, and let V be, in some sense, small compared to H0. Then it is expected that the spectral properties of H are close to those of H0. Typically, H0 is simpler than H, and in many cases we know its spectral family E0(·) explicitly. The task of perturbation theory is to deduce information about the spectral properties of H = H0 + V from those of H0. We shall always consider the case of self-adjoint operators on a Hilbert space H. The spectrum of a self-adjoint operator has two components: discrete (i.e., eigenvalues) and continuous. Hence, perturbation theory has two parts: perturbation theory for the discrete spectrum, and for the continuous spectrum. Eigenvalues of H0 can generically be shifted under arbitrary small perturbations, but the formulas for these shifts are basically the same as in the finite dimensional case, dim H < ∞, which is linear algebra. On the contrary, the continuous spectrum is much more stable. Example. Let ∞ H0 = −∆,H = −∆ + v(x), v(x) = v(x), v ∈ L (R), (1.1) be self-adjoint operators on the Hilbert space H = L∈(R). Denote by Φ the Fourier transform. Then ∗ 2 H0 = Φ |ξ| Φ, (1.2) 2 so that the spectrum of H0 is the same as that of the multiplication operator |ξ| . Moreover, we know all functions of the operator H0 and, in particular, its spectral family explicitly. In the case considered V is multiplication by the function v. Thus, each of the operators H0 and V is very simple. However, it is not quite so simple to understand their sum, the Schrödinger operator H = H0 + V . Nevertheless, it is easy to deduce from the Weyl theorem that, if v(x) → 0 as |x| → ∞, then 1 2 DMITRI YAFAEV the essential spectrum of H is the same as that of H0 and hence it coincides with [0, ∞). Scattering theory requires classification of the spectrum in terms of the theory of measure. Each measure may be decomposed into three parts: an absolutely continuous part, a singular continuous part, and a pure point part. The same classification is valid for the spectral measure E(·) of a self-adjoint operator H. √√ Thus, there is a decomposition of the Hilbert space H = Hac ⊕ H∫c ⊕ H into the orthogonal sum of invariant subspaces of the operator H; the operator restricted to √√ Hac, H∫c or H shall be denoted Hac, Hsc or Hpp, respectively. The pure point part corresponds to eigenvalues. The singular continuous part is typically absent. Actually, a part of scattering theory is devoted to proving this for various operators of interest, but in these lectures we study only the absolutely continuous part Hac of H. The same objects for the operator H0 will be labelled by the index ‘0’. We ac denote P0 the orthogonal projection onto the absolutely continuous subspace H0 of H0. The starting point of scattering theory is that the absolutely continuous part of self-adjoint operator is stable under fairly general perturbations. However assumptions on perturbations are much more restrictive than those required for stability of the essential spectrum. So scattering theory can be defined as perturbation theory for the absolutely continuous spectrum. Of course, it is too much to expect that ac ac H = H0 . However, we can hope for a unitary equivalence: ac ac ∗ ac ac H = UH0 U ,U : H0 → H onto. The first task of scattering theory is to show this unitary equivalence. We now ask: how does one find such a unitary equivalence U? This is related (although the relationship is not at all obvious) to the second task of scattering theory, namely, the large time behaviour of solutions u(t) = e−iHtf. of the time-dependent equation ∂u i = Hu, u(0) = f ∈ H. ∂t If f is an eigenvector, Hf = λf, then u(t) = e−iλtf, so the time behaviour is evi- dent. By contrast, if f ∈ Hac, one cannot, in general, calculate u(t) explicitly, but scattering theory allows us to find its asymptotics as t → ±∞. In the perturbation theory setting, it is natural to understand the asymptotics of u in terms of solutions of the unperturbed equation, iut = H0u. It turns out that, under rather general ac ± ac assumptions, for all f ∈ H , there are f0 ∈ H0 such that

± ± −iH0t ± u(t) ∼ u0 (t), t → ±∞, where u0 (t) = e f0 , or, to put it differently, −iHt −iH t ± lim e f − e 0 f = 0. (1.3) t→±∞ 0 ± Hence f0 and f are related by the equality f = lim eiHte−iH0tf ±, t→±∞ 0 which justifies the following fundamental definition given by C. Møller [23] and made precise by K. Friedrichs [8]. LECTURES ON SCATTERING THEORY 3

Definition 1.1. The limit

iHt −iH0t W± = W±(H,H0) = s- lim e e P0, t→±∞ if it exists, is called the wave operator. ± It follows that f = W±f0 . The wave operator has the properties ac (i) W± is isometric on H0 . (ii) W±H0 = HW± (the intertwining property). ac In particular, H0 is unitarily equivalent, via W±, to the restriction H|ran W± of ac H on the range ran W± of the wave operator W± and hence ran W± ⊂ H . ac Definition 1.2. If ran W± = H , then W± is said to be complete.

It is a simple result that W±(H,H0) is complete if and only if the ‘inverse’ wave operator W±(H0,H) exists. Thus, if W± exists and is complete (at least for one ac ac of the signs), H0 and H are unitarily equivalent. It should be emphasized that scattering theory is interested only in the canonical unitary equivalence provided by the wave operators. Another important object of scattering theory ∗ S = W+W− (1.4) is called the scattering operator. It commutes with H0, SH0 = H0S, which follows ac directly from property (ii) of the wave operators. Moreover, it is unitary on H0 if both W± are complete. In the spectral representation of the operator H0, the operator S acts as multiplication by the operator-valued function S(λ) known as the scattering matrix (see the next lecture, for more details). An important generalization of Definition 1.1 is due to Kato [19]. Definition 1.3. Let J be a bounded operator. Then the modified wave operator W±(H,H0,J) is defined by

iHt −iH0t W±(H,H0,J) = s- lim e Je P0, (1.5) t→±∞ when this limit exists. Modiﬁed wave operators still enjoy the intertwining property

W±(H,H0,J)H0 = HW±(H,H0,J), ac but of course their isometricity on H0 can be lost.

2. We have seen that the wave operators give non-trivial spectral information about H. Thus, it is an important problem to find conditions guaranteeing the existence of wave operators. There are two quite different methods: the trace class method, and the smooth method (see the next lecture). The trace class method is the principal method of abstract scattering theory. For applications to differential operators, both methods are important. The fundamental theorem for the trace class method is the Kato-Rosenblum theorem [15, 31, 16]. Recall that a compact operator T on H is in the class Sp, p > 0, if p X ∗ p/2 ||T ||p = λj(T T ) < ∞.

In particular, S1 is called the trace class and S2 is called the Hilbert-Schmidt class. 4 DMITRI YAFAEV

Theorem 1.4. If the difference V = H − H0 belongs to the trace class, then the wave operators W±(H,H0) exist. This is a beautiful theorem. It has a number of advantages, including: (i) Since the conditions are symmetric with respect to the operators H0 and H, the wave operators W±(H0,H) also exist and hence W±(H,H0) are complete. (ii) The formulation is simple, but all proofs of it are rather complicated. (iii) It relates very different sorts of mathematical objects: operator ideals, and scattering theory. (iv) It is effective, since it is usually easy to determine whether V is trace class. (v) It is sharp, in the sense that if H0 and p > 1 are given, there is a V ∈ Sp such that the spectrum of H0 + V is purely point. However, Theorem 1.4 has a disadvantage: it is useless in applications to differential operators. Indeed, for example, for the pair (1.1) V is a multiplication operator which cannot be even compact (unless identically zero), and therefore Theorem 1.4 does not work. Nevertheless, it is still useful if one is only interested ac ac in the unitary equivalence of H and H0 = H0 . In fact, it is sufficient to show −1 −1 that the operators (H +c) and (H0 +c) are unitarily equivalent for some c > 0 or, according to Theorem 1.4, that their difference is trace class. Actually, this condition is sufficient for the existence (and completeness) of the wave operators. More generally, the following result is true. Theorem 1.5. Suppose that −n −n (H − z) − (H0 − z) ∈ S1 for some n = 1, 2,... and all z with Im z 6= 0. Then the wave operators W±(H,H0) exist and are complete. Theorem 1.5 was proved in [3] for n = 1 and in [17] for arbitrary n. For semi- bounded operators, Theorem 1.5 follows from the Invariance Principle, due to Bir- man [1].

Theorem 1.6. Suppose that ϕ(H) − ϕ(H0) ∈ S1 for a real function ϕ such that its derivative ϕ0 is absolutely continuous and ϕ0(λ) > 0. Then the wave operators W±(H,H0) exist and

W±(H,H0) = W±(ϕ(H), ϕ(H0)). 0 The operators W±(H,H0) should be replaced here by W∓(H,H0) if ϕ is negative.

3. A typical result of trace class theory in applications to diﬀerential operators is the following 2 Theorem 1.7. Let H0 = −∆ + v0(x) and H = −∆ + v0(x) + v(x) on L (R). Assume that v0 =v ¯0 is bounded and v =v ¯ satisﬁes |v(x)| ≤ Chxi−ρ, hxi = (1 + |x|2)1/2, (1.6) for some ρ > d. Then the wave operators W±(H,H0) exist and are complete. For simplicity we shall give the proof only for d = 1, 2 or 3. We proceed from Theorem 1.5 for n = 1. Choose c > 0 large enough so that H0 + c and H + c are invertible. Then −1 −1 −1 −1 (H + c) − (H0 + c) = −(H + c) V (H0 + c) . LECTURES ON SCATTERING THEORY 5

Denote temporarily H00 = −∆. Since the operators −1 −1 (H + c) (H00 + c) = Id −(H + c) (V0 + V ) −1 and, similarly, (H00 + c)(H0 + c) are bounded, it suﬃces to check that

−1 −1 −1 1/2 1/2 −1 (H00 + c) V (H00 + c) = (H00 + c) |V | sgn v |V | (H00 + c) ∈ S1. (1.7)

−1 1/2 Note that Φ(H00 + c) |V | is an integral operator with kernel (2π)−d/2e−ix·ξ(|ξ|2 + c)−1|v(x)|1/2, 2 −1 1/2 which is evidently in L (R2). Thus, the operators (H00 +c) |V | and its adjoint 1/2 −1 |V | (H00 + c) are Hilbert-Schmidt and hence (1.7) holds. Using Theorem 1.5 for n > d/2 − 1, it is easy (see [4], for details) to extend this result to an arbitrary d. On the contrary, the condition ρ > d in (1.6) cannot be improved in the trace-class framework.

4. Theorem 1.4 admits the following generalization (see [26]) to the wave operators (1.5).

Theorem 1.8. Suppose that V = HJ − JH0 ∈ S1. Then the wave operators W±(H,H0; J) exist. This result due to Pearson allows to simplify considerably the original proof of Theorem 1.4 and of its diﬀerent generalizations. Although still rather sophisti- cated, the proof of Theorem 1.8 relies only on the following elementary lemma of Rosenblum. Lemma 1.9. For a self-adjoint operator H, consider the set R ⊂ Hac of elements f such that 2 rH (f) := ess-sup d(E(λ)f, f)/dλ < ∞. If G is a Hilbert-Schmidt operator, then for any f ∈ R Z ∞ 2 2 2 ||G exp(−iHt)f|| dt ≤ 2πrH (f)||G||2. −∞ Moreover, the set R is dense in Hac. The wave operators were deﬁned in terms of exponentials, e−itH . There is also a ‘stationary’ approach, in which the exponentials are replaced by the resolvents R(z) = (H −z)−1, z = λ±iε, and the limit t → ±∞ is replaced by the limit ε → 0. In the trace class framework a consistent stationary approach was developed in the paper [2]. From analytical point of view it relies on the following result on boundary values of resolvents which is interesting in its own sake.

Proposition 1.10. Let H be a self-adjoint operator and let G1, G2 be arbitrary Hilbert-Schmidt operators. Then the operator-function G1R(λ+iε)G2 has limits as ε → 0 (and G1R(z)G2 has angular boundary values as z → λ ± i0) in the Hilbert- Schmidt class for almost all λ ∈ R. Moreover, the operator-function G1E(λ)G2 is diﬀerentiable in the trace norm for almost all λ ∈ R. In particular, Proposition 1.10 allows one to obtain a stationary proof of the Pearson theorem (see the book [33]). 6 DMITRI YAFAEV

2. Smooth method. Short range scattering theory 1. The smooth method relies on a certain regularity of the perturbation in the spectral representation of the operator H0. There are different ways to understand regularity. For example, in the Friedrichs-Faddeev model [7] H0 acts as multiplication by independent variable in the space H = L2(Λ; N) where Λ is an interval and N is an auxiliary Hilbert space. The perturbation V is an integral operator with sufficiently smooth kernel. Another possibility is to use the concept of H-smoothness introduced by T. Kato in [18]. Definition 2.1. Let G be an H-bounded operator; that is, suppose that G(H +i)−1 is bounded. Then we say that G is H-smooth if there is a C < ∞ such that Z ∞ kGe−iHtfk2dt ≤ C2kfk2 (2.1) −∞ for all f ∈ H or, equivalently, sup kGR(z) − R(z)G∗k ≤ 2πC2. (2.2) Im z6=0 In applications the assumption of H-smoothness of an operator G imposes too stringent conditions on the operator H. In particular, the operator H is necessarily absolutely continuous if kernel of G is trivial. This excludes eigenvalues and other singular points in the spectrum of H, for example, the bottom of the continuous spectrum for the Schrödinger operator with decaying potential or edges of bands if the spectrum has the band structure. However it is often suffices to verify H- smoothness of the operators GE(Xn) where the union of intervals Xn exhausts R up to a set of the Lebesgue measure zero. In this case we say that G is locally H-smooth. We have the following

∗ Theorem 2.2. Let H −H0 = G G0, where G0 is locally H0-smooth and G is locally H-smooth. Then the wave operators W±(H,H0) exist and are complete. This is a very useful theorem, yet the proof is totally elementary. To make it even more simple, we forget about the word ‘locally’ and assume that G0 is H0-smooth and G is H-smooth. We write Z t iHt −tH0t −sH0 −iHs lim e e f0, f = f0, f + lim i G0e f0, Ge f ds. t→±∞ t→±∞ 0 Since the left and right hand sides of the last inner product are L2(R), the limit on the right hand side exists. This shows existence of the weak limit. To show the strong limit, we estimate using (2.1) t 0 0 Z iHt −tH0t iHt −tH0t −sH0 −iHs e e f0, f − e e f0, f = G0e f0, Ge f ds t0 Z t 1/2 −sH0 2 ≤ Ckfk kG0e f0k ds . t0 Taking the sup over f with kfk = 1, we obtain t 0 0 Z 1/2 iHt −tH0t iHt −tH0t −sH0 2 ke e f0 − e e f0k ≤ C kG0e f0k ds t0 LECTURES ON SCATTERING THEORY 7 which goes to zero as t0, t tend to infinity. Of course Theorem 2.2 is not effective since the verification of H0 - and espe- cially of H-smoothness may be a difficult problem. In the following assertion the −1 hypothesis only concerns the free resolvent, R0(z) = (H0 − z) . Theorem 2.3. Suppose V can be written in the form V = G∗ΩG, where Ω is a ∗ bounded operator and GR0(z)G is compact for Im z 6= 0, and is norm-continuous up to the real axis except possibly at a finite number of points λk, k = 1,...,N. Then W±(H,H0) exist and are complete. We give only a brief sketch of its proof. Set N [ −1 −1 Xn = (−n, n) \ (ak − n , ak + n ). k=1 ∗ Since kGR0(λ ± iε)G k is uniformly bounded for λ ∈ Xn, the operator GE0(Xn) is H0-smooth (cf. the definition (2.2)). To show a similar result for the operator-function GR(z)G∗, we use the resolvent identity

R(z) = R0(z) − R0(z)VR(z), (2.3) whence −1 ∗ ∗ ∗ GR(z)G = Id +GR0(z)G Ω GR0(z)G , Im z 6= 0. (2.4) It easily follows from self-adjointness of the operator H that the homogeneous equation ∗ f + GR0(z)G Ωf = 0 (2.5) ∗ has only the trivial solution f = 0 for Im z 6= 0. Since the operator GR0(z)G is compact, this implies that the inverse operator in (2.4) exists. By virtue of equation (2.4), the operator-function GR(z)G∗ is continuous up to the real axis except the points λk and the set N of λ such that the equation (2.5) has a non-trivial solution for z = λ + i0 or z = λ − i0. The set N is obviously closed. Moreover, it has the Lebesgue measure zero by the analytical Fredholm alternative. This implies that the pair G, H also satisﬁes the conditions of Theorem 2.2. It remains to use this theorem.

2. Let us return to the Schrödinger operator H = −∆+v. Potentials v satisfying (1.6) with ρ > 1 are said to be short range. Below in this lecture we make this assumption. Let us apply Theorem 2.3 to the pair H0 = −∆, H. Put r = ρ/2 −r ∗ and G = hxi . One can verify (cf. the proof of Theorem 1.7) that GR0(z)G is compact for Im z 6= 0 (and arbitrary r > 0). Next we consider the spectral family E0(λ) which according to (1.2) satisfies 2 d(E0(λ)f, f)/dλ = ||Γ0(λ)f|| , (2.6) where √ −1/2 (d−2)/4 ˆ (Γ0(λ)f)(ω) = 2 λ f( λω) (2.7) and fˆ = Φf is the Fourier transform of a function f from, say, the Schwartz class. ˆ Thus, up to a numerical factor, Γ0(λ)f is the restriction of f to the sphere of radius 8 DMITRI YAFAEV √ λ. Remark further that if f ∈ hxi−rL2(R), then fˆ belongs to the Sobolev space Hr(R). Since r > 1/2, it follows from the Sobolev trace theorem that the operator −r 2 − Γ0(λ)hxi : L (R) → L2(S 1) is bounded and depends (in the operator norm) Hölder continuously on λ > 0. Therefore according to (2.6) the operator-function GE0(λ)G is differentiable and its derivative is also Höldercontinuous. Now we use the representation Z ∞ −1 (R0(z)Gf, Gf) = (λ − z) d(E0(λ)Gf, Gf), 0 which, by the Privalov theorem, implies that the operator-function GR0(z)G is continuous in the closed complex plane cut along [0, ∞) except, possibly, the point z = 0. Applying Theorem 2.3, we now obtain

Theorem 2.4. Let v be short range. Then the wave opeartors W±(H,H0) for the pair (1.1) exist and are complete.

ac ± Theorem 2.4 implies that for every f ∈ H , there is f0 such that relation (1.3) holds. Using the well-known expression for kernel of the integral operator e−itH0 (in the x-representation), we ﬁnd that −itH i|x|2/4t −d/2 ˆ± (e f)(x) ∼ e (2it) f0 (x/2t). (2.8) Here ‘∼’ means that the diﬀerence of the left and right hand sides tends to zero in L2(Rd) as t → ±∞. Thus, the solution ‘lives’ in the region |x| ∼ |t| of (x, t) space. As a by-product of our considerations we obtain that the operator-function GR(z)G is continuous up to real axis, except a closed set of measure zero. More detailed analysis shows that this set consists of eigenvalues of the operator H (and possibly the point zero), so that the singular continuous spectrum of H is empty. Finally, we note that, by the Kato theorem, the operator H does not have positive eigenvalues. This gives the following assertion known as the limiting absorption principle. Theorem 2.5. Let v be short range and r > 1/2. Then the operator-function hxi−rR(z)hxi−r is norm-continuous in the closed complex plane cut along [0, ∞) except negative eigenvalues of the operator H and, possibly, the point z = 0.

3. Let us compare Theorems 1.7 and 2.4. If v0 = 0, then Theorem 2.4 is stronger because, in assumption (1.6), it requires that ρ > 1 whereas Theorem 1.7 requires that ρ > d. Theorem 2.4 can be extended to some other cases, for example to periodic and long-range v0. In the ﬁrst case the spectral family E0(·) can be constructed rather explicitly. In the second case the limiting absorption principle can be also veriﬁed (see the next lecture). However, contrary to Theorem 1.7, the ∞ method of proof of Theorem 2.4 gives nothing for arbitrary v0 ∈ L . Therefore the following question naturally arises.

Problem. Let H0 = −∆ + v0(x) and H = −∆ + v0(x) + v(x) where v satisﬁes estimate (1.6) for ρ > 1. Do the wave operators W±(H,H0) exist for an arbitrary ∞ v0 ∈ L ? This problem is of course related to uniﬁcation of trace class and smooth approaches. The following theorem does this to some extent: LECTURES ON SCATTERING THEORY 9

Theorem 2.6. Assume that V is of the form V = G∗ΩG, where Ω is a bounded ∗ ∗ operator, GR0(z)G ∈ Sp for some p < ∞, and GR0(z)G has angular boundary values in Sp for almost every λ ∈ R. Then W±(H,H0) exist and are complete. Note that, given Proposition 1.10, Theorem 2.6 provides an independent proof of Theorem 1.4. On the other hand, it resembles Theorem 2.3 of the smooth approach. However it gives nothing for the solution of the problem formulated above. Most probably, the answer to the formulated question is negative, which can be considered as a strong evidence that a real uniﬁcation of trace class and smooth approaches does not exist.

4. The stationary method is intimately related to eigenfunction expansions of the operators H0 and H. Let us discuss this relation on the example of the Schr¨odinger operator H with a short range potential. For the operator H0 = −∆,√ the con- i λω·x struction of eigenfunctions is obvious. Actually, if ψ0(x, ω, λ) = e , then −∆ψ0 = λψ0. This collection of eigenfunctions is ‘complete’, so eigenfunctions are parametrized by ω ∈ Sd−1 (for ﬁxed λ > 0). By the intertwining property, the wave operators W±(H,H0) diagonalize H and hence

∗ 2 ∗ HW±Φ = |ξ| W±Φ . Thus, at least formally, eigenfunctions of H, that is, solutions of the equation

−∆ψ + vψ = λψ, (2.9) can be constructed by one of the two equalities ψ+(ω, λ) = W+ψ0(ω, λ) or ψ−(ω, λ) = W−ψ0(ω, λ). It turns out that this definition can be given a precise sense, and one can construct solutions of the Schrödinger equation with asymptotics ψ0(x, ω, λ) at infinity. Theorem 2.7. Assume (1.6) is valid for some ρ > d. For every λ > 0 and ω ∈ Sd−1 there is a solution ψ of (2.9) such that √ √ ψ(x, ω, λ) = ei λω·x + a(ˆx, ω, λ)|x|−(d−1)/2ei λ|x| + o(|x|−(d−1)/2), xˆ = x/|x|, (2.10) where a is a continuous function on Sd−1 × Sd−1. √ √ We interpret the ei λω·x term as an incoming plane wave, the |x|−(d−1)/2ei λ|x| term as an outgoing spherical wave, and the coefficient a(ˆx, ω, λ) is called the scattering amplitude for the incident direction ω and the direction of observationx ˆ. Note that a solution of the Schrödinger equation is determined uniquely by the condition that, asymptotically, it is a sum of a plane and the outgoing spherical waves. Under assumption (1.6) where ρ > (d + 1)/2, eigenfunctions of the operator H may be constructed by means of the following formula:

ψ(ω, λ) = ψ0(ω, λ) − R(λ + i0)V ψ0(ω, λ), (2.11) or, equivalently, as solutions of the Lippman-Schwinger equation

ψ(ω, λ) = ψ0(ω, λ) + R0(λ + i0)V ψ(ω, λ). 10 DMITRI YAFAEV

We note that, strictly speaking, the second term in the right hand side of (2.11) is deﬁned by the equality

−1/2− (d+1)/2+2 −d/2− R(λ + i0)V ψ0(ω, λ) = R(λ + i0)hxi (V hxi )(hxi ψ0(ω, λ)), (2.12)

−d/2− 2 d where ε > 0 is suﬃciently small. Here hxi ψ0(λ, ω) ∈ L (R ) and the operator V hxi(d+1)/2+2 is bounded. Therefore, by the limiting absorption principle (The- orem 2.5), the function (2.12) belongs to the space hxi1/2+L2(Rd) for any > 0. The asymptotics (2.10) is still true for arbitrary ρ > (d + 1)/2 if the remainder e(x) = o(|x|−(d−1)/2) is understood in the following averaged sense: Z −(d−1)/2 1 2 e(x) = oav(|x| ) ⇐⇒ lim |e(x)| dx = 0. R→∞ R |x|≤R

In this case the scattering amplitude a(ˆx, ω, λ) belongs to the space L2(Sd−1) in d−1 the variablex ˆ uniformly in ω ∈ S . √ It is often convenient to write ψ in terms of the parameter ξ = λω ∈ R, instead of (ω, λ). Thus, we set

ψ−(x, ξ) = ψ(x, ω, λ), ψ+(x, ξ) = ψ−(x, −ξ). The Schwartz kernels of the wave operators are intimately related to eigenfunctions. In fact, if we define ‘distorted Fourier transforms’ Φ± by Z −d/2 (Φ±f)(ξ) = (2π) ψ±(x, ξ)f(x)dx, d R 2 so that Φ±H = |ξ| Φ±, then we have ∗ W± = Φ±Φ0, (2.13) where Φ0 = Φ is the classical Fourier transform. Notice that the right hand side is defined purely in terms of time-independent quantities. This can be taken to be the definition of the wave operators in the stationary approach to scattering theory. Historically, it was the first approach to the study of wave operators suggested by Povzner in [27, 28] and developed further in [10]. Under optimal assumption ρ > 1 in (1.6) Theorem 2.4 was obtained in [21]. It follows from (2.13) that the scattering operator

∗ ∗ ∗ S = W+W− = Φ0Φ+Φ−Φ0. (2.14)

Let H0 be realized (via the Fourier transform) as multiplication by λ in the space 2 + − L (R ; L2(S 1)). Since S commutes with H0 and is unitary, it acts in this representation as multiplication by the operator-function (scattering matrix) S(λ): L2(Sd−1) → L2(Sd−1) which is also unitary for all λ > 0. It can be deduced from (2.14) that Z (S(λ)f)(ω) = f(ω) + γc(λ) a(ω, ω0, λ)f(ω0)dω0, (2.15) d−1 S where a is the scattering amplitude deﬁned by (2.10) and

γ = eπi(d−3)/4, c(λ) = i(2π)−(d−1)/2λ(d−1)/4. LECTURES ON SCATTERING THEORY 11

If assumption (1.6) is satisfied for ρ > 1 only, then the scattering matrix satisfies the relation ∗ S(λ) = Id −2πiΓ0(λ) V − VR(λ + i0)V Γ0(λ), (2.16) −r −r which generalizes (2.15). Here Γ0(λ) is the operator (2.7). Since hxi V hxi is a bounded operator for r = ρ/2 > 1/2, we see that (2.16) is correctly defined. It follows from formula (2.16) that the operator S(λ) − Id is compact. Thus, the spectrum of S(λ) (which lies on the unit circle by unitarity) is discrete, and may accumulate only at the point 1. Moreover, if α −ρ−|α| ∂ v(x) ≤ Cαhxi (2.17) for all multi-indices α (and ρ > 1), then the kernel k(ω, ω0) of the operator S(λ)−Id is smooth for ω 6= ω0 and |k(ω, ω0)| ≤ C|ω − ω0|−d+ρ. For ρ ∈ (1, (d + 1)/2], construction of eigenfunctions, which behave asymptotically as plane waves, becomes a difficult problem. In particular, formula (2.11) makes no sense in this case. One can do something, but the construction of [32] is rather complicated and requires the condition (2.17). On the contrary, for arbitrary ρ > 1 we can construct solutions which formally correspond to averaging of ψ(x, ω, λ) over ω ∈ Sd−1. To illustrate this idea, let us first consider the free case v = 0. Then, for any b ∈ C∞(Sd−1), the function Z √ u(x) = ei λω·xb(ω)dω d−1 S satisfies −∆u = λu and, by the stationary phase arguments, it has asymptotics √ √ u(x) = c(λ)−1|x|−(d−1)/2 γb¯ (ˆx)ei λ|x| − γb(−xˆ)e−i λ|x| + o(|x|−(d−1)/2). In the general case one can also construct solutions of the Schrödinger equation (2.9) with the asymptotics of incoming and outgoing spherical waves. Theorem 2.8. Assume (1.6) is valid for some ρ > 1. Let u be a solution of the Schrödingerequation (2.9) satisfying Z |u(x)|2 dx ≤ CR, for all R ≥ 1. |x|≤R 2 d−1 Then there are b± ∈ L (S ) such that √ √ −(d−1)/2 i λ|x| −i λ|x| −(d−1)/2 u(x) = |x| γb¯ +(ˆx)e − γb−(−xˆ)e + oav(|x| ). (2.18)

Functions b± are related by the scattering matrix : b+ = S(λ)b−. Conversely, for 2 d−1 2 d−1 2 d−1 all b+ ∈ L (S )(or b− ∈ L (S )), there is a unique function b− ∈ L (S ) 2 d−1 (or b+ ∈ L (S )), and a unique solution u of (2.9) satisfying (2.18). 3. Long range scattering theory There are different and, to a large extent, independent methods in long range scattering (see [36]). Here we shall give a brief presentation of the approach of the paper [34] which relies on the theory of smooth perturbations. 1. The condition (1.6) with ρ > 1 is optimal even for the existence of wave operators for the pair H0 = −∆, H = −∆ + v(x). For example, the wave operators −1 do not exist if v(x) = v0hxi , v0 6= 0. Nevertheless the asymptotic behaviour of 12 DMITRI YAFAEV the function exp(−iHt)f for large |t| remains sufficiently close to the free evolution exp(−iH0t)f0 if the condition (2.17) is satisfied for |α| ≤ α0 with α0 big enough. Potentials obeying this condition for some ρ ∈ (0, 1] are called long-range. There are several possible descriptions of exp(−iHt)f as t → ±∞. One of them is a modification of the free evolution which, in its turn, can be done either in momentum or in coordinate representations. Here we discuss the coordinate modification. Motivated by (2.8), we set

−d/2 ˆ (U0(t)f)(x) = exp(iΞ(x, t))(2it) f(x/(2t)), (3.1) where the choice of the phase function Ξ depends on v. Then the wave operators are deﬁned by the equality

iHt W± = s- lim e U0(t). (3.2) t→±∞ To be more precise, these limits exist if the function Ξ(x, t) is a (perhaps, approximate) solution of the eikonal equation ∂Ξ/∂t + |∇Ξ|2 + v = 0. For example, if ρ > 1/2, we can neglect here the nonlinear term |∇Ξ|2 and set Z 1 Ξ(x, t) = (4t)−1|x|2 − t v(sx)ds. (3.3) 0 In the general case one obtains an approximate solution of the eikonal equation by the method of successive approximations. With the phase Ξ(x, t) constructed in ˆ ∞ d such a way, for an arbitrary f ∈ C0 (R \{0}), the function U0(t)f is an approximate solution of the time-dependent Schr¨odinger equation in the sense that Z ±∞

||(i∂/∂t − H)U0(t)f||dt < ∞. 1 iHt 1 This condition implies that the vector-function ∂e U0(t)f/∂t ∈ L (R) and hence the limit (3.2) exists. The modified wave operators have all the properties of usual ac wave operators. They are isometric, W±H0 = HW± and ran W± ⊂ H . As in ac the short-range case, the wave operator is said to be complete if ran W± = H . Only the completeness of W± is a non-trivial mathematical problem. As we shall see below, it has a positive solution which implies that for every f ∈ Hac, there is ± f0 such that (cf. (2.8)) −itH iΞ(x,t) −d/2 ˆ± (e f)(x) ∼ e (2it) f0 (x/2t). (3.4) Thus, in the short and long range cases, the large time asymptotics of solutions of the time-dependent Schrödinger equation differ only by the phase factor. In particular, in both cases they live in the region |x| ∼ |t| of (x, t) space.

2. For the proof of completeness of wave operators, we need some analytical results which we discuss now. Note ﬁrst of all that the limiting absorption principle, Theorem 2.5, remains true if assumption (2.17) is satisﬁed for some ρ > 0 and |α| ≤ 1. However perturbative arguments of the previous lecture do not work for long range potentials. The simplest proof (see the original paper [24] or the book LECTURES ON SCATTERING THEORY 13

[6]) of the limiting absorption principle relies in this case on the Mourre estimate: d X iE(I)[H,A]E(I) ≥ cE(I),A = −i (∂jxj + xj∂j), c > 0, (3.5) j=1 where I is a sufficiently small interval about a given λ > 0. This is not too hard to prove for long-range two body potentials (but is much harder for multiparticle operators). To be more precise, the Mourre estimate implies that the operator-function hxi−rR(z)hxi−r is norm-continuous up to positive half-axis, except a discrete set of eigenvalues of the operator H. In particular, the operator hxi−r is locally H- smooth which is sufficient for scattering theory. Moreover, independent arguments (see, e.g., [30]) show that the Schrödinger operator H does not have positive eigenvalues. Unfortunately, the operator hxi−1/2 is not (locally) H-smooth even in the case v = 0. However, there is a substitute: Theorem 3.1. Let assumption (2.17) be satisfied for some ρ > 0 and |α| ≤ 1. Set ⊥ −2 (∇j u)(x) = (∂ju)(x) − |x| ((∇u)(x) · x)xj, j = 1, . . . , d. −1/2 ⊥ Then the operators hxi ∇j E(Λ) are H-smooth for any compact Λ ⊂ (0, ∞). The proof is based on the equality d −1 X ⊥ 2 −1−ρ 2|x| |∇j u| = ([H, ∂r]u, u) + (v1u, u), v1(x) = O(|x| ), j=1 which is obtained by direct calculation. Since hxi−(1+ρ)/2E(Λ) is H-smooth, we only have to consider the term [H, ∂r]. For this we note that Z t −isH −isH −itH −itH i [H, ∂r]e f, e f ds = (∂re f, e f) − (∂rf, f). 0 For f ∈ ran E(Λ), the right hand side is bounded by C(Λ)kfk2, so this shows −1/2 ⊥ H-smoothness of the operators hxi ∇j E(Λ). Note that the commutator method used for the proof of Theorem 3.1 goes back to Putnam [29] and Kato [20].

3. Our proof of completeness relies on consideration of wave opeartors (1.5) with a specially chosen operator J. Let us look for J in the form of a pseudodiﬀerential operator Z (Jf)(x) = (2π)−d/2 eix·ξp(x, ξ)fˆ(ξ)dξ. (3.6) d R m ∞ We shall work in the class of symbols Sρ,δ consisting of functions p(x, ξ) ∈ C (R × R) satisfying the estimates α β m−ρ|α|+δ|β| ∂x ∂ξ p(x, ξ) ≤ Cα,βhxi for all multi-indices α and β. In addition, we shall assume that p vanishes for |ξ| ≥ R, for some R. The number m is called the order of the symbol, and of the corresponding pseudodiﬀerential operator. Compared to the usual calculus, here the roles of x and ξ are interchanged: we require some decay estimates as |x| → ∞, rather than as |ξ| → ∞ in the usual situation. We shall assume that 14 DMITRI YAFAEV

0 ≤ δ < 1/2 < ρ ≤ 1. The symbol p(x, ξ) of the operator J belongs to the class 0 2 d Sρ,δ. We recall that operators of order zero are bounded on L (R ). Since iHt −iH0t iHt −iH0t de Je /dt = ie HJ − JH0)e , it is desirable to find such J that the effective perturbation HJ − JH0 be short- range (a pseudodifferential operator of order −1 − ε for ε > 0). This means that ψ(x, ξ) = eix·ξp(x, ξ) is an approximate eigenfunction of the operator H, with eigenvalue |ξ|2, for each ξ. Let us look for ψ in the form ψ(x, ξ) = eiφ(x,ξ). Then (−∆ + v(x) − |ξ|2)ψ = (|∇φ|2 + v(x) − |ξ|2 − i∆φ)ψ, which leads to the eikonal equation |∇φ|2 + v(x) = |ξ|2. Suppose that φ(x, ξ) = x · ξ + Φ(x, ξ), (3.7) where α β 1−ρ−|α| ∂x ∂ξ Φ(x, ξ) = O(|x| ). (3.8) Then |∇φ|2 + v(x) − |ξ|2 = 2ξ · ∇Φ + |∇Φ|2 + v(x) (3.9) and neglecting the nonlinear term we obtain the equation

2ξ · ∇xΦ + v(x) = 0. (3.10) We need its two solutions given by the equalities 1 Z ∞ Φ±(x, ξ) = ± v(x ± tξ) − v(±tξ) dt. (3.11) 2 0

Clearly, these functions Φ± satisfy assumption (3.8) but only off a conic neighbourhood of x = ∓ξ. Thus, two problems arise. The first is that, to obtain in (3.6) a symbol p± 0 from the class Sρ,δ, we need to remove by a cut-off function ζ±(x, ξ) a small conic neighbourhood of the setx ˆ = ∓ξˆ. Thus, we are obliged to consider the wave operators for two different operators J±. This idea appeared in [11] and will be realized below. The second problem is that the term |∇Φ|2 neglected in (3.9) is ‘short range’ only for ρ > 1/2. Of course, it is easy to solve the eikonal equation by iterations, considering at each step a linear equation of type (3.10), and to obtain a ‘short range’ error for arbitrary ρ > 0. However, even after the cut-off by the 0 function ζ±(x, ξ), we obtain the symbol p± from the class Sρ,1−ρ, which is bad if ρ ≤ 1/2. To overcome this difficulty, we need to take into account the oscillating nature of p± (see [35]).

∞ 4. Below we suppose that ρ > 1/2. Let σ± ∈ C be such that σ±(θ) = 1 near ±1 and σ±(θ) = 0 near ∓1. We construct J± by the formula (3.6) where

iΦ±(x,ξ) p±(x, ξ) = e ζ±(x, ξ) (3.12) ˆ and the cut-off function ζ±(x, ξ) essentially coincides with σ±(hx,ˆ ξi). We deliber- ately ignore here some technical details which can be found in [34]. For example, strictly speaking, additional cut-offs of low and high energies by a function of |ξ|2 LECTURES ON SCATTERING THEORY 15 and of a neighbourhood of x = 0 by a function of x should be added to ζ±(x, ξ). Then the operators J± so constructed are pseudodifferential operators of order 0 and type (ρ, δ = 1 − ρ). Note that Theorem 2.2 extends automatically to the wave operators (1.5). Thus, we are looking for a factorization ∗ HJ± − J±H0 = G ΩG, where G is locally H0- and H-smooth and the operator Ω is bounded. Let us recall that the operator hxi−r is H-smooth, for any r > 1/2. Since zeroth order pseudodifferential operators are bounded on L2(Rd), a factorization as above would have been true if HJ − JH0 were of order −1 − for some > 0. However, the pseudodifferential operator HJ± −J±H0 has symbol decaying only −1 ix·ξ as |x| ; this is from one derivative of −∆ hitting e and one hitting ζ±. Notice that the symbol of the ‘bad’ term equals

iΦ±(x,ξ) ˆ −2ie hξ, ∇xσ±(hx,ˆ ξi)i. It only decays as |x|−1 but is supported outside a conic neighbourhood of the set wherex ˆ = ξˆ orx ˆ = −ξˆ. Therefore, d X −1/2 ⊥ ∗ −1/2 ⊥ −r −r HJ − JH0 = (hxi ∇j ) Ωj(hxi ∇j ) + hxi Ω0hxi , r > 1/2, j=1 where Ωj, j = 0, 1, . . . , d, are order zero pseudodiﬀerential operators. Using Theo- rem 3.1 for the ﬁrst d terms, and the limiting absorption principle for the last one, we obtain

∗ Proposition 3.2. The wave operators W±(H,H0,Jτ ) and W±(H0,H,Jτ ) exist for τ = ‘+’ and τ = ‘−’. The next step is to verify

Proposition 3.3. The operators W±(H,H0; J±) are isometric and W±(H,H0; J∓) = 0. Indeed, it suﬃces to check that

∗ −iH0t s- lim (J±J± − I)e = 0 (3.13) t→±∞ and

∗ −iH0t s- lim J∓J∓e = 0. (3.14) t→±∞ ∗ According to (3.12), up to a compact term, J∓J∓ equals the pseudodiﬀerential 2 operator Q∓ with symbol ζ∓(x, ξ). If t → ±∞, then the stationary point ξ = x/(2t) of the integral

Z 2 −iH0t −d/2 ihξ,xi−i|ξ| t 2 ˆ (Q∓e f)(x) = (2π) e ζ∓(x, ξ)f(ξ)dξ (3.15) d R 2 does not belong to the support of the function ζ∓(x, ξ). Therefore supposing that d f ∈ S(R ) and integrating by parts, we estimate integral (3.15) by CN (1 + |x| + |t|)−N for an arbitrary N. This proves (3.14). To check (3.13), we apply the same 2 arguments to the PDO with symbol ζ±(x, ξ) − 1. Now it is easy to prove the asymptotic completeness. 16 DMITRI YAFAEV

Theorem 3.4. Suppose that condition (2.17) is fulﬁlled for ρ > 1/2 and all α. Then the wave operators W±(H,H0; J±) exist, are isometric and complete. ∗ ∗ Since W±(H0,H; J∓) = W±(H,H0; J∓), it follows from Proposition 3.3 that ∗ W±(H0,H; J∓) = 0. This implies that ∗ −iHt ac lim ||J∓e f|| = 0, f ∈ H . (3.16) t→±∞ 2 2 Let us choose the functions σ± in such a way that σ+(θ) + σ−(θ) = 1. Then ∗ ∗ J+J+ + J−J− = Id +K for a compact operator K and hence ∗ −iHt 2 ∗ −iHt 2 2 ||J+e f|| + ||J−e f|| = ||f|| + o(1) as |t| → ∞. Now it follows from (3.16) that ∗ −iHt lim ||J±e f|| = ||f||. t→±∞ This is equivalent to isometricity of the wave operators ∗ ∗ W±(H0,H; J±) = W±(H,H0; J±), so that W±(H,H0; J±) are complete. We emphasize that Theorem 3.4 and, essentially, its proof remain valid for an arbitrary ρ > 0. Now it easy to justify the asymptotics (3.4). By existence and completeness of the wave operator W±(H,H0,J±), we have

−itH −itH0 ± ± ∗ lim ke f − J±e f k = 0, f = W±(H0,H,J±)f. t→±∞ 0 0 The critical point ξ(x, t) of the integral

Z 2 −itH0 ± −d/2 ix·ξ iΦ±(x,ξ) −it|ξ| ˆ± (J±e f0 )(x) = (2π) e e ζ±(x, ξ)e f0 (ξ) dξ d R (3.17) is deﬁned by the equation

2tξ(x, t) = x + ∇ξΦ±(x, ξ(x, t)), ±t > 0, so that ξ(x, t) = x/(2t)+O(|t|−ρ). Applying stationary phase to the integral (3.17), we obtain formula (3.4) with function 2 Ξ(x, t) = |x| /(4t) + Φ±(x, x/(2t)), which equals (3.3).

5. In the long range case the scattering operator S is defined again by formula (1.4) where W± = W±(H,H0,J±). Thus, again S is unitary and commutes with H0, so defines a family of scattering matrices S(λ) which are unitary operators on L2(Sd−1) for λ > 0. However the structure of the spectrum of the scattering matrix in the short and long range cases are completely different. Typically, in the long range case the spectrum of S(λ) covers the unit circle. The nature of this spectrum is in general not known, except in the radially symmetric case (that is, when the potential v is a function only of |x|). In that case, the scattering matrix commutes with the Laplacian on Sd−1, so it breaks up into the orthogonal sum of finite dimensional operators. Hence the spectrum is dense pure point in this case. LECTURES ON SCATTERING THEORY 17

The kernel s(ω, ω0, λ) of S(λ) is still smooth away from the diagonal, but near the diagonal it, typically, has the form (for ρ < 1)

−1 0 s(ω, ω0, λ) ∼ c(ω, λ)|ω − ω0|−(d−1)(1+ρ )/2eiθ(ω,ω ,λ), (3.18) where θ(ω, ω0, λ) is asymptotically homogeneous of order 1 − ρ−1 as ω − ω0 → 0. In the long range case practically nothing is known about eigenfunctions, which behave asymptotically as plane waves. Moreover, as shows the explicit formula (see, −1 e.g., [22]) for the Coulomb potential v(x) = v0|x| , in this case the separation of the asymptotics of eigenfunctions into the sum of plane and spherical waves loses, to a large extent, its sense. On the contrary, Theorem 2.8 extends [9] to arbitrary long range potentials.

4. The scattering matrix. High energy and smoothness asymptotics This section relies on the paper [37]. 1. Let us begin with the short range case, (1.6) with ρ > 1. Then there is a stationary representation (2.16) of the scattering matrix. Using the Sobolev trace theorem and the dilation transformation x 7→ λ−1/2x, we can show (see [36], for details) that, for any r > 1/2, the operator (2.7) satisﬁes the estimate −r −1/4 kΓ0(λ)hxi k ≤ C(r)λ . Similarly, one can control the dependence on λ in the limiting absorption principle, which yields khxi−rR(λ + i0)hxi−rk ≤ C(r)λ−1/2. The representation (2.16) allows us to study by a simple iterative procedure the behaviour of the scattering matrix S(λ) in the two asymptotic regimes of interest, namely for high energies and in smoothness of its kernel. In fact these two regimes are closely related. Namely, we use the resolvent identity (2.3) and substitute its right hand side in place of the resolvent R(λ + i0) in (2.16). This gives us a series known as the Born approximation:

N n X n ∗ S(λ) = Id −2πi (−1) Γ0(λ)V R0(λ + i0)V Γ0(λ) + σN (λ). n=0 −(N−2)/2 The error term, σN (λ) is O(λ ) in operator norm, and in addition gets smoother and smoother in the sense that σN (λ) ∈ Sα(N), where αN → 0 as N → ∞. However, the Born approximation has several drawbacks. First, it is actually quite complicated, involving multiple oscillating integrals of higher and higher di- mensions as n increases. Second, it does not apply to long range potentials, or to magnetic, even short range, potentials. Here we discuss another, much simpler, form of approximation which applies to all these situations but requires assumptions of the type (2.17).

∞ 2. To give an idea of the approach, suppose first for simplicity that v ∈ C0 (R) (though the argument can be applied to a wider class of potentials). Let the solution of the Schrödinger equation (2.9) be defined by formula (2.11). By (2.16), the integral kernel k(ω, ω0, λ) of the operator S(λ) − Id may be written as Z √ k(ω, ω0, λ) = −iπ(2π)−dλ(d−2)/2 e−i λx·ωv(x)ψ(x, ω0, λ) dx. (4.1) d R 18 DMITRI YAFAEV

Since this is an integral over a compact region, to analyze the asymptotics of k(ω, ω0, λ) as λ → ∞, it suﬃces to construct the asymptotics of ψ(x, ω, λ) for bounded x as λ → ∞. This can be done by the following well known procedure [5]. Assuming for a moment only (2.17) for ρ > 1, one seeks ψ(x, ω, λ) in the form N √ ix·ξ X −n ˆ ψ(x, ξ) = e b(x, ξ), b(x, ξ) = (2i|ξ|) bn(x, ξ), b0 = 1, ξ = λω. n=0 (4.2) Plugging (4.2) into the Schr¨odinger equation (2.9), and equating powers of |ξ|, we obtain equations ˆ ξ · ∇xbn+1 = −∆bn + vbn. (4.3) The remainder term −ix·ξ 2 rN (x, ξ) = e (−∆ + v(x) − |ξ| )ψ(x, ξ) is then given by −N ˆ ˆ rN (x, ξ) = (2i|ξ|) − ∆bN (x, ξ) + v(x)bN (x, ξ) . Equations (4.3) can be explicitly solved: Z 0 ˆ ˆ ˆ ˆ ˆ ˆ bn+1(x, ξ) = − ∆bn(x + tξ, ξ) + v(x + tξ)bn(x + tξ, ξ) dt. −∞ It is easy to see that α β ˆ −(ρ−1)n−|α| −|β| ∂x ∂ bn(x, ξ) ≤ Cα,βhxi |ξ| , ξ (4.4) α β −1−(ρ−1)(N+1)−|α| −N−|β| ∂x ∂ξ rN (x, ξ) ≤ Cα,βhxi |ξ| , except on arbitrary conic neighbourhoods of the bad directionx ˆ = ξˆ. If v is compactly supported, then the estimates in x are inessential, so it follows from (4.1) and (4.2) that

N √ Z √ 0 −d (d−2)/2 X −n i λx·(ω0−ω) 0 k(ω, ω , λ) = −iπ(2π) λ (2i λ) e v(x)bn(x, ω ) dx d n=0 R +O(λ(d−3)/2−N/2). Since N is arbitrary, this gives the asymptotic expansion of the scattering amplitude as λ → ∞. We note that k ∈ C∞(Sd−1 × Sd−1) in the variables ω and ω0, so the ∞ d smoothness asymptotics in the case v ∈ C0 (R ) is trivial.

3. Finally, we give a universal formula which applies in both the long range and magnetic cases. We shall need two approximate solutions ψ± of the Schr¨odinger equation 2 2 i∇ + a(x) ψ± + v(x)ψ± = |ξ| ψ±. We suppose that a vector (or magnetic) potential a(x) as well as scalar potential v(x) satisfy the condition (2.17) for some ρ > 0. Let us look for ψ± in the form

iφ±(x,ξ) ψ±(x, ξ) = e b±(x, ξ), (4.5) LECTURES ON SCATTERING THEORY 19 where φ = φ± is defined by formula (3.7) and Φ = Φ± satisfies (3.8). Plugging (4.5) into the Schrödinger equation, we obtain the eikonal equation for φ: 2 2 2 |∇xφ| − 2a(x) · ∇xφ + v0(x) = |ξ| , v0(x) = |a(x)| + v(x). (4.6)

If ρ > 1 and a = 0, then one can set Φ = 0. In this case ψ−(x, ξ) = ψ(x, ξ) and ψ+(x, ξ) = ψ(x, −ξ) where the function ψ(x, ξ) was constructed in the previous subsection. However, even if a is short range (but does not vanish), then, for the study of the limit λ → ∞, one cannot avoid the eikonal equation. Once again, the equation (4.6) for function φ = φ± deﬁned by formula (3.7) where Φ = Φ± can be solved by successive approximations:

N0 X −n ˆ Φ(x, ξ) = (2|ξ|) φn(x, ξ). n=0 Here ˆ ˆ ξ · ∇φ0 + ξ · a = 0, ˆ 2 ξ · ∇φ1 + |∇φ0| − 2a · ∇φ0 + v0 = 0 n ˆ X ξ · ∇φn+1 + ∇φm · ∇φn−m − 2a · ∇φn = 0, n ≥ 1. m=0 So at every step we have an equation ˆ ˆ ˆ ξ · ∇xφ(x, ξ) + f(x, ξ) = 0, with, possibly a long range function f. This equation can be solved by one of the two formulas (cf. (3.10), (3.11)) Z ∞ ˆ ˆ ˆ ˆ φ±(x, ξ) = ± f(x ± tξ, ξ) − f(±tξ, ξ) dt. (4.7) 0

Using both signs ‘+’ and ‘−’, we obtain functions φ± satisfying (4.6) up to a term q±(x, ξ) such that

α β −N0ρ−|α| −N0−|β| ∂x ∂ξ q±(x, ξ) ≤ Cα,βhxi |ξ| for all (x, ξ) excluding an arbitrary conic neighbourhood of the directionx ˆ = ξˆ for ˆ the minus sign, orx ˆ = −ξ for the plus sign. One chooses and ﬁxes N0 such that N0ρ > 1. Then for b± one has the transport equation

−2iξ · ∇b± + 2i(a − ∇Φ±) · ∇b± − ∆b± + (−i∆Φ± + idiv a + q±)b± = 0. (4.8)

(N) As before, one looks for b± = b± in the form (4.2) which gives standard equations ˆ (±) ˆ (±) ˆ ξ · ∇xbn+1(x, ξ) = fn (x, ξ), (±) (±) (±) where fn are determined by functions b1 , . . . , bn . All these equations are (±) solved by formula (4.7) (but the term fn (±tξˆ) can be dropped). Then the func- (±) (±) tions bn and the remainder rN in the transport equation satisfy estimates of the type (4.4) for some ρ > 1. As N → ∞, we obtain a function (4.5) satisfying the Schr¨odinger equation with an arbitrary given accuracy both in the variables x and ξ. 20 DMITRI YAFAEV

Now we can give an approximate formula for the scattering amplitude. Remark that away from the diagonal, it is not hard to show that the kernel s(ω, ω0, λ) of the scattering matrix S(λ) is smooth and O(λ−∞), so the main point is to understand 0 d−1 the kernel when ω and ω are close to some given point ω0. Let ω0 ∈ S be arbitrary, let Πω0 be the plane orthogonal to ω0 and x = zω0 + y, y ∈ Πω0 . Set Z √ √ 0 (d−2)/2 −d 0 s0(ω, ω , λ) = ∓πiλ (2π) ψ+(y, λω)∂zψ−(y, λω ) Πω0 √ √ √ √ ! 0 0 (4.9) −∂zψ+(y, λω)ψ−(y, λω ) − 2i(a(y) · ω0)ψ+(y, λω)ψ−(y, λω ) dy.

(N) Here s0 = s0 depends on N, since the construction above depends on a choice of N. On the contrary, it does not depend on N0 which is fixed. Let Ω± = Ω±(ω0, δ) ⊂ d−1 S be determined by the condition ±ω · ω0 > δ > 0 and Ω = Ω+ ∪ Ω−. Then we have Theorem 4.1. For all integer p, there is an N = N(p) such that (N) 0 0 (N) 0 p s˜0 (ω, ω , λ) := s(ω, ω , λ) − s0 (ω, ω , λ) ∈ C (Ω × Ω) and (N) −p s˜0 (·, ·, λ) Cp ≤ Cλ . We make some brief comments on the proof of this theorem. Note that the representation formula (2.16) holds in the general short range case, when both electric and magnetic potentials are present. However it is useless for the proof of Theorem 4.1 even for purely electric short range potentials. As in the previous lecture, one considers instead modified wave operators W±(H, H0,J±), where Z −d/2 ˆ (J±f)(x) = (2π) ψ±(x, ξ)ζ±(x, ξ)f(ξ) dξ d R and the functions ψ±(x, ξ) are defined by formula (4.5). Compared to the previous lecture, there are two important differences in the construction of the operators J±. First, we cannot neglect high energies and therefore have to control the dependence on ξ in all estimates. Second, for the proof of the existence and completeness of the wave operators, it was sufficient to take b± = 1 in (4.5). On the contrary, now (N) b± = b± is an approximate solution of the transport equation (4.8) depending on the parameter N, which allows us to obtain an arbitrary good approximation (N) ψ± = ψ± to solutions of the Schrödinger equation. Writing T± for the effective perturbation,

T± = HJ± − J±H0, we have (see [12, 34, 36]) S(λ) = S0(λ) + S1(λ), where ∗ ∗ S0(λ) = −2πiΓ0(λ)J+T−Γ0(λ), ∗ ∗ (4.10) S1(λ) = 2πiΓ0(λ)T+R(λ + i0)T−Γ0(λ).

Note that T± is a pseudodiﬀerential operator of order −1, so that precise meaning of (4.10) needs to be explained. The special reason why S0 is correctly deﬁned is that LECTURES ON SCATTERING THEORY 21

∗ the amplitude of the pseudodiﬀerential operator J+T− is zero in a neighbourhood ˆ ˆ of the set wherex ˆ is close to ξ or −ξ or, to put√ it diﬀerently, in a neighbourhood of the conormal bundle to every sphere |ξ| = λ. The term S1(λ) turns out to be negligible for large N. Theorem 4.2. For all integer p, there is an N = N(p) such that (N) 0 p d−1 d−1 s1 (ω, ω , λ) ∈ C (S × S ), and (N) −p s1 (·, ·, λ) Cp ≤ Cλ . The proof relies on propagation estimates (see [25, 14, 13]) following from the Mourre estimate (3.5). We give an example of such an estimate. Let again A be the generator of dilations. Then for all integers p p p −1 hxi EA(R−)R(λ + i0)EA(R+)hxi = O(λ ).

Thus, the interesting part of the scattering matrix is contained in the term S0(λ) of (4.10). It is very explicit representation, but has a drawback because it depends on the cutoﬀs ζ±. So one has to transform the expression for S0(λ) to the invariant expression (4.9), which does not contain the cutoﬀs ζ±. This is the least obvious part of the proof of Theorem 4.1. Finally, we note that formula (3.18) for the diagonal singularity of the scattering amplitude can be obtained applying the stationary phase method to integral (4.9).

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Department of Mathematics, University of Rennes – I, Campus de Beaulieu 35042 Rennes, FRANCE E-mail address: [email protected]